How to make this texture use displacement modifier?
If you want circles, i'd suggest to use the Voronoi Texture in combination with a Color Ramp.

To get the offset from the border you will need to use the mapping node with 0.5 offset in x and y. You can play around with the scale of the voronoi texture, but there are only certain value ranges where you get the circle shapes. Important that "Randomness" is set to 0!

Also regarding the highly subdivided geometry, you can use the "Experimental" Cycles setup, where you can use adaptive subdivision.





The picture above should be self explanatory, but important to note is that you will have to use the "Experimental" Cycles Engine setup to gain access to the adaptive subdivision as shown below:



Here you need to adjust the dicing scale to get a smoother result.
In the "Surface" panel option in the material, as shown below, you can set the "Displacement Only" or "Displacement and Bump".



You can always fall back to using Bump only which does not need the "Experimental" Setting and should be a lot less strain on the system. But that would depend on how close/detailed you want to go.